JEE Advanced 2018 · Paper 1

For a non-zero complex number z, let arg(z) denote the principal argument with −π < arg(z) ≤ π. Then, which of the following statement(s) is (are) FALSE?

1. Aarg(−1 − i) = π/4, where i = √−1
2. BThe function f: ℝ → (−π, π], defined by f(t) = arg(−1 + it) for all t ∈ ℝ, is continuous at all points of ℝ, where i = √−1
3. CFor any two non-zero complex numbers z₁ and z₂, arg(z₁/z₂) − arg(z₁) + arg(z₂) is an integer multiple of 2π
4. DFor any three given distinct complex numbers z₁, z₂ and z₃, the locus of the point z satisfying the condition arg(((z−z₁)(z₂−z₃))/((z−z₃)(z₂−z₁))) = π, lies on a straight line

In a triangle PQR, let ∠PQR = 30° and the sides PQ and QR have lengths 10√3 and 10, respectively. Then, which of the following statement(s) is (are) TRUE?

1. A∠QPR = 45°
2. BThe area of the triangle PQR is 25√3 and ∠QRP = 120°
3. CThe radius of the incircle of the triangle PQR is 10√3 − 15
4. DThe area of the circumcircle of the triangle PQR is 100π

For every twice differentiable function f: ℝ → [−2, 2] with (f(0))² + (f′(0))² = 85, which of the following statement(s) is (are) TRUE?

1. AThere exist r, s ∈ ℝ, where r < s, such that f is one-one on the open interval (r, s)
2. BThere exists x₀ ∈ (−4, 0) such that |f′(x₀)| ≤ 1
3. Climₓ→∞ f(x) = 1
4. DThere exists α ∈ (−4, 4) such that f(α) + f″(α) = 0 and f′(α) ≠ 0

Let f: ℝ → ℝ and g: ℝ → ℝ be two non-constant differentiable functions. If f′(x) = (e^(f(x)−g(x)))g′(x) for all x ∈ ℝ, and f(1) = g(2) = 1, then which of the following statement(s) is (are) TRUE?

1. Af(2) < 1 − logₑ 2
2. Bf(2) > 1 − logₑ 2
3. Cg(1) > 1 − logₑ 2
4. Dg(1) < 1 − logₑ 2